You are here probably because you do not remember these formulas. Your brain was not built to remember meaningless formulas. Keep reading to understand these formulas (or how to learn math).

Area of a Circle

Draw a circle. Now draw a square just surrounding the circle. The key things is, that percentage is the same no matter what the size of the circle. That percentage is about 78.5%.

The line from one side of the circle to the other is called the diameter. It is usually abreviated as d. So

• the length of the side of the square is also d
• the area of the square is d squared (d times d), and

the area of the circle is .785 times d squared

Wasn't that easy?

Circumference

The circumference of the circle -- the line around the outside of the circle -- can be calculated in the same way. The circumference of the circle is just a percentage of the circumference (perimeter) of the square, and it's the same percentage no matter what the size.

Believe it or not, that percentage is 78.5% again. (Isn't that incredible? A seemingly random number is the answer to two different problems.)

• The sides of the square are d
• the perimeter of the square is 4 * d (because there are four sides)

So the formula for the circumference of a circle is .785 * 4 * d.

Mathematicians and Pi

Maybe you would want to give this ratio, 78.5%, a name. Maybe pie would be a good name, because a circle looks like a pie. Then the formula for the area of a circle is pie times d squared.

If your teacher wants an approximate numerical answer, you can use 78.5%. But what if your teacher wants the exact answer? To be exact, you would have to use pie in your answer. As you know, you shouldn't be turning in homework with pie on it.

The good news is, mathematicians have already done the work for you. The bad news, not only did they misspell the name, the gave it to the wrong value. It's called pi, and it's four times that value of pie. In other words, pi = 4 * pie and pie = pi/4. That works out to be about 3.14.

Why did they do this? Well, they were in a hurry and didn't think closely about it. And they weren't psychologists. Actually, when you use pi, you can express the formulas a little more simply. Our formulas were

area = d * d * pie
circumference = 4 * d * pie.

Subsituting pi/4 for pie, and then simplifying by using the radius (r) rather than the diameter, these formulas become:

area = d * d * pi / 4      =     r * r * pi
circumference = d * pi

Voila! There you have it. Two simple formulas, using pi instead of pie. Never mind that you won't remember the two formulas. Never mind that all rationale for using pi is completely lost. Never mind that the formulas don't make sense. Mathematicians!

Bob Sez

This website is about teaching math through insight, which teaches mental models, understand, and concepts. Not memorization (which you forget).